Tag: mathematics

Posted in Science & Nature

Card Shuffling

Posted on by Jinavie

Take a standard deck of playing cards. Shuffle it thoroughly and set it on the table. Consider this: what is the probability that the order those 52 cards are in is the same as the order of a deck shuffled by someone else? The answer can be found using a simple maths equation: 52!

! denotes a factorial, where you multiply the number to every other positive integer smaller than it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. Due to its nature, factorials grow rapidly – even faster than exponentials. For example, 10! is 3.6 million and 15! is 1.3 quadrillion. By 52!, the number grows to:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000.

This number is so big that if every star in our galaxy had a trillion planets, each with a trillion people living on it, all shuffling a trillion deck of cards at the rate of 1000 shuffles per second, since the beginning of time, only now would someone have a deck that is in the exact order as your deck.

Ergo, you can say with absolute, mathematical certainty, that the deck you have shuffled is in an order never created by any human being in the history of the world.

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Posted in Philosophy

The First Move

Posted on by Jinavie

In the game of chess, every move counts. Each action you take can drastically change the way the game will play out from there on. By the second move, there are 72,084 possible games. By the third move, 9 million possible games exist. By the fourth, there are 318 trillion possible games. Essentially, after the first move, the game becomes nearly impossible to predict. There are more possible games on a chessboard than there are atoms in the universe. What spawns all of these possibilities is the first move.

This makes the first move all that terrifying. One mistake and you have destroyed countless possibilities where you are victorious. On the last few moves of the game, the results are much more predictable as the possibilities have been weeded out. Therefore, you can have more confidence in your moves. But the first move is as far as you can get from the end move, with an infinite sea of possibilities between you and the other side.

The corollary to this is that if you do make a mistake on your first move, then you have infinite ways to fix that mistake. So don’t be afraid of taking the first move – simply relax and play the game.

(inspiration/paraphrased from Harold Finch, Person of Interest)

Posted in History & Literature

Maps

Posted on by Jinavie

Think of the world map. Most of you will think of the typical map where Europe and Africa are in the middle, with Russia dominating the Eurasian landmass and Greenland easily outsizing South America.

Mercator projection

The most common world map we use nowadays is based on the Mercator projection. Because the Earth is spherical and maps are two-dimensional rectangles, complex mathematics are involved to project the former on the latter by distorting the picture. The Mercator projection was created by Flemish cartographer Gerardus Mercator in 1569. The map was extremely useful for sailors because it depicted the curvature of the Earth in straight lines, making navigation much easier.

However, the Mercator projection severely distorts the size of each continents, meaning the image of the world we have in our heads is completely misleading. According to the Mercator projection, Greenland is as large as Africa, Alaska is as large as continental USA and Antarctica dwarfs every continent.

Gall-Peters projection

To solve this problem, the Gall-Peters projection was suggested in 1974 as an alternative as it correctly displayed the continents’ respective sizes. As you can see, in reality Greenland is significantly smaller than even Australia, Europe and Russia are much smaller than expected and Africa is an extremely large continent.

Dr. Arno Peters argued that the Mercator projection was a biased, euro-centric projection that harmed the world’s perception of developing countries. This of course, led to extreme controversy over the politics of cartography.

Africa vs Greenland

There are many other distortions commonly found in maps. Maps tend to enlarge the landmass of the own country subtly and some American maps go as far as placing the USA in the middle even if it means splitting Eurasia in half. Even though landmass does not correlate in any way with how well the lives of its inhabitants are, such distortions can be seen even nowadays. This shows that not everything you see is as it seems.

Posted in Science & Nature

Addition

Posted on by Jinavie

In the late 18th century, the great mathematician Carl Friedrich Gauss was given a punishment by his teacher for being mischievous. The punishment was this simple yet tedious problem: add every integer number from 1 to a 100.

Gauss, now referred to as the “Princeps mathematicorum” (Latin for “the Prince of Mathematicians”), came up with a simple shortcut and solved the problem without breaking a sweat. He realised that he could add two numbers from opposite ends of the range of numbers and get the same number e.g. 1 + 100 = 101, 2 + 99 = 101 etc. Using this logic, there must be a certain number of identical pairs of 101.

He then came up with the following equation:

100/2 x (1 + 100) = 50 x 101 = 5050

Posted in History & Literature

Baker’s Dozen

Posted on by Jinavie

A dozen is a counting term used to describe 12 of something. But when you have a baker’s dozen of bread, you have 13 pieces of baking, not 12. This may seem like a charitable gift from the baker, but the historical origin is somewhat different.

In the Middle Ages (particulary around the 13th century), baking was not an exact science and loaves of breads were made with varying sizes and weights. This made it easy for bakers to short the customer by giving them smaller loaves than what the customer needed.

To stop this, many countries implemented laws that prevented bakers from shorting the customer, usually by setting a minimum weight for a dozen loaves of bread. However, it is entirely possible for the baker to lose a few loaves of breads to accidental dropping, burning or thieves stealing them. Because the breads may come out smaller, it could not be guaranteed that a dozen loaves would be heavy enough to meet the guidelines – no matter how honest the baker was. To offset this, bakers began adding an extra loaf to ensure that they would not disobey the law (and pay a hefty fine or be seriously punished).

Another theory with less historical evidence is based on the shape of baking trays. Most baking trays are made in a 3:2 ratio and the most efficient way to place loaves of breads on these trays is a 4:5:4 hexagonal arrangement. This arrangement has the advantage of avoiding the corners, where the temperature will heat up then cool down faster, making the results less perfect. Therefore, bakers may have sold a batch of 13 loaves together instead of selling 12 and leaving one out.

Posted in Science & Nature

Interesting Numbers

Posted on by Jinavie

One of the more humorous sides to numbers is mathematicians’ attempts to categorise numbers as “interesting” or “dull”. For example, 1 is interesting because it is the first positive integer. 73 is interesting because it is the 21st prime number and 21 is a multiple of 7 and 3. The number 1729 is a good example of how a number can seem dull but later found to be interesting. When the British mathematician G. H. Hardy visited Indian mathematician Srinivasa Ramanujan, he commented that the number of the taxicab he rode in on was 1729 – a number he found to be rather dull. Ramanujan objected and stated it is very interesting as it is the smallest number expressible as the sum of two cubes in two different ways (1729 = 1³ + 12³ = 9³ + 10³). Such numbers are now referred to as taxicab numbers and 1729 is called the Hardy-Ramanujan number.

A way to discover the smallest most uninteresting number is through the Online Encyclopaedia of Integer Sequences, which documents every integer worth noting as it is in some sort of arithmetic sequence. The smallest integer that does not appear in this encyclopaedia as part of a sequence could be considered as objectively the smallest “uninteresting” number. In 2009, this number was 11630, but has since changed to 12407, then 13794 and now 14228 (22 April 2014).

But paradoxically, the smallest uninteresting number is interesting in itself by being the smallest most uninteresting number. This is known as the interesting number paradox. By this paradox, every natural number is unique and ergo, “interesting”.

Number Line

(Image source: http://www.xkcd.com/899/, and here’s an explanation of some of the numbers on it)

Posted in Science & Nature

Pigeonhole Principle

Posted on by Jinavie

If I have three gloves, there must be at least either two left gloves or two right gloves. It is impossible to have one left glove, one right glove and a third glove that is neither left nor right (usually). This logic is called the pigeonhole principle. It is named because of the logic that if you have n pigeons and m pigeonholes where n > m (e.g. 10 pigeons in 9 holes), then at least one pigeonhole must contain more than one pigeon. This is because the biggest spread of the pigeons is putting at least one in each box, but as n > m, there is a pigeon left over and it must go in a box with another pigeon. The pigeonhole principle seems like a basic counting principle, but its implications are quite interesting.

For example, let’s say that your sock drawer is very unorganised and has a mix of black and white socks. What is the minimum number of socks you need to pick out before you get two of the same colour? The pigeonhole principle dictates that when n > m, each “slot” must be filled with more than one item. Here, the slot is colour. As there are two colours (m = 2), you only have to pick three socks out to have a matching pair (n = 3, 3 > 2).

The pigeonhole principle allows us to make seemingly impossible conjectures, such as the fact that a person living in London will have the exact number of hairs on their head as at least one other person living in London. An average human head has about 150,000 hairs and it would be a safe assumption to say that no one would have more than a million hairs on their head (m = 1,000,000). The population of London far exceeds a million (n > 1,000,000), therefore, there must at least two people living in London with the exact same amount of hair on their head. Similarly, if you are in a room with 366 other people, you are guaranteed to share a birthday with at least one person.

Posted in Science & Nature

Millions And Billions

Posted on by Jinavie

Have you ever stopped and pondered what a million actually is? Sure, you might easily pass it off as the number 1,000,000, or a thousand thousands, but have you really tried to get your head around how big a number that is? For example, you may be able to visualise a hundred people, a thousand people or even tens of thousands of people in your head, but it is very hard to visualise an image of a million people.

Now consider this. When was a million seconds ago? You know a second is very short and a million is a very large number. But it is difficult to put the two together. Make a guess. Last year? Two months ago? Surprisingly, the answer is only a week and a half ago (11.6 days).
Then what about a billion seconds? A billion is a thousand million so you might think it is easy to just add some zeroes, but a billion seconds is 31.7 years ago. Just by changing one syllable, or adding three zeroes, we went from a scale of weeks to years. If we go one step further to a trillion seconds, you leap back in time 31,700 years. You can probably remember what happened a million seconds ago, you might not have even been born a billion seconds ago and our ancestors were still hunter-gatherers roaming Europe a trillion seconds ago. That is how mind-blowing the scale of large numbers can be.

Now let’s look at some other things to really understand how big a million and a billion can be. A million dollars (USD) could buy you a luxury house, a manufacturing line, a 41-acre island in Belize or over 200 years’ worth of coffee (if you drank two cups a day). A million dollars in $1 bills would weigh 1000kg and stack to 30 stories high. A billion dollars – even if you were to convert it into $100 bills – would weigh 10 tonnes, almost as heavy as the truck that would carry it.

The pitter-patter of raindrops on your face feels nice, but a million drops of water weighs 50kg and would break your neck. A billion red helium balloons would have enough lift to carry 14,000 tonnes – enough to lift a hundred small, two-storey houses up into the air. A million grains of rice will feed a person for almost two months, while a billion ants would weigh twice a standard car (3 tonnes total).

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(You should definitely check out Hank Green’s take on “a million seconds”, because everything is better if Hank Green is ranting about it! http://www.youtube.com/watch?v=cJ7A0yTDiqQ)

Posted in Life & Happiness

The Perfect Toast

Posted on by Jinavie

Toast is one of those simple meals that anyone can make. Bread goes in, toast comes out. But some scientists decided to embark on a quest for the “perfect” toast. After spending a week toasting and tasting over two thousand slices of toast, the scientists came up with some figures.

The perfect toast should be:

  • 14mm thick
  • Made from pale-seeded loaf of bread taken from a fridge at 3°C
  • Cooked in a 900-watt toaster set to 5 out of 6 power
  • Cooked at a temperature of 154°C evenly from both sides
  • Cooked for exactly 3 minutes and 36 seconds (216 seconds)
  • Transferred gently to a plate that is pre-warmed to 45°C
  • Immediately slathered with 68.2mg per square centimetre of butter
  • Sliced once diagonally

The result of this formula is a perfectly golden-brown toast of 12:1 exterior to interior crispiness, with the “ultimate balance of external crunch and internal softness”.

Posted in Science & Nature

Around The World

Posted on by Jinavie

Imagine, if you will, a very long piece of ropethat loops around the Earth, fitting it tightly around the equator like a belt. If you wanted to raise this rope off the surface by one metre all around, how much more rope will you need?

The length of rope is the same as the circumference of the Earth which is 40,075km (24,901 miles). Ergo, it is easy to think that you would need kilometres of rope to extend it enough to float a metre off the Earth’s surface. However, in reality you need a mere 6.28m of extra rope to achieve this.

The reason is extremely simple, mathematically speaking. The circumference of any given circle is given by the equation 2πr, where r is the radius of the circle. Therefore, if you increase r by 1 unit (e.g. 1m), then the circumference increases by 2π x 1 = 2π = 6.28. No matter how large the circle may be, this rule does not change.

(This is a famous maths riddle, but here’s a much more interesting application of the concept in this What If? article. God I love that blog! http://what-if.xkcd.com/67/)